Curved surface modeling method based on boundary information

Abstract

The invention provides a curved surface modeling method based on boundary information. The method comprises the following steps: inputting a peripheral curved surface of an automobile windshield; generating boundary information control points according to boundary geometric continuity splicing conditions; generating internal control points of the curved surface by adopting a homogeneous biharmonic equation model; and integrating the boundary control points and the internal control points of the intermediate curved surface, and generating the intermediate curved surface according to an NURBS curved surface generation algorithm. According to the method, the problems of high calculation complexity, difficulty in boundary matching and the like of a traditional method for realizing boundary information-based curved surface modeling are effectively solved, and an efficient and accurate curved surface modeling method is provided.

Description

Technical Field

[0001] The present invention belongs to the field of complex geometric shape design, and particularly relates to a surface modeling method based on boundary information. Background Art

[0002] In the field of industrial design, generating surface models that meet design requirements based on boundary information has a wide range of application scenarios. For example, in the field of automobile manufacturing, by precisely controlling the shape of the boundary curves, smooth surfaces that meet the design requirements can be generated for parts such as windshields, hoods, doors, and trunks. This not only achieves smooth lines and good aerodynamic performance but also enhances the aesthetic value of the automobile's appearance and reduces wind resistance. In the design of electronic product casings, generating casings for electronic products such as mobile phones and tablets with simple and smooth surface models based on boundary information can help designers quickly create high-quality surface models for product casings, meeting the aesthetic standards and ergonomic requirements of the products.

[0003] Traditional surface modeling methods based on boundary information, such as the Coons surface patch construction method, mainly rely on fine-tuning control points to generate the desired surface. Although these methods perform well in dealing with simple geometric shapes, when faced with complex geometric models, it is difficult for the intermediate surfaces to meet the design standards. To ensure good integration of the intermediate surfaces with the surrounding surfaces, methods based on partial differential equations (PDEs) have gradually become an effective way to solve this problem. Compared with traditional surface modeling methods based on boundary information, PDE methods control surface modeling through parameters, boundary conditions, or the right-hand side term. Due to the smoothness of their solutions, they can construct smoother surfaces of higher quality than traditional methods. This method introduces fewer parameters, simplifies the shape control process, and is suitable for dealing with complex geometries. The traditional method for generating surface models based on PDEs designs algorithms with the surface itself as the solution of the PDE. This design method requires different PDEs for processing different geometric continuous boundary information, and at the same time, the algorithm requires a large amount of computation to reach a given geometric error threshold. Summary of the Invention

[0004] The object of the present invention is to provide a surface modeling method based on boundary information. On the basis of accurately satisfying the peripheral surface boundary information of the automotive windshield, a naturally transitioning surface is generated, thereby effectively solving problems such as high computational complexity and difficult boundary matching existing in the traditional surface modeling method based on boundary information. G1 continuity means that the surface is not only position - continuous but also tangential - continuous at the boundary. Here, according to the peripheral surface of the automotive windshield and the requirements of G1 continuity, the boundary control points of the intermediate surface (i.e., the surface where the windshield is located, hereinafter simply referred to as the intermediate surface) are directly calculated and generated. For the internal control points, the homogeneous bi - harmonic equation is solved by the difference method to obtain them. Finally, the boundary control points and the internal control points are integrated, the control points of the intermediate surface are output, and according to the defined knot vector, the intermediate surface is generated.

[0005] The technical solution for achieving the object of the present invention is as follows:

[0006] A surface modeling method based on boundary information, comprising:

[0007] Step 1, input the peripheral surface of the automotive windshield: Given the control grid and the corresponding weights of the peripheral surface of the automotive windshield, construct a NURBS surface as the peripheral surface;

[0008] Step 2, calculate the boundary information: According to the fact that the peripheral surface and the intermediate surface are G 1 -continuous, let and

[0009] Thus, the outermost control points and the corresponding weights of the intermediate surface and the sub - outermost control points and the corresponding weights of the intermediate surface at the splicing position are obtained;

[0010] where R(0, v) and respectively represent the control points corresponding to when the parameter u takes zero and the parameter v takes the corresponding value in the surface patch R(u, v) and the surface patch ; R(u, v) is a surface patch on the peripheral surface with u, v as parameters, is a surface patch on the intermediate surface with u, v as parameters;

[0011] Step 3, calculate the internal control points of the intermediate surface: Use the discrete format based on the bi - harmonic equation to construct and solve the system of equations AP * = G, obtain the vector P * , and extract the points included in the solution region Ω in the vector P * , and integrate them into the vector P. Each component of the vector P is the internal control point of the intermediate surface;

[0012] where A is a matrix constructed according to the difference format, P * is the grid point (xs , y t ) corresponding unknown quantity P s,t a vector formed by, G is a vector formed by integrating the outer - layer control points and newly - defined points, and arranging these points in the order from left to right and from top to bottom. The newly - defined points are the points where all three coordinate components are set to 0 within the rectangular grid formed by the outer - layer control points;

[0013] Step 4, output the complete surface: Integrate the boundary control points obtained in Step 2 and the internal control points obtained in Step 3 to form a complete control - point matrix. According to the defined knot vector, use the NURBS surface generation algorithm to generate the intermediate surface, that is, the surface corresponding to the automotive windshield.

[0014] Compared with the existing methods, the remarkable advantages of the present invention are:

[0015] (1) High boundary - information matching degree: The present invention uses the homogeneous bi - harmonic equation model for surface modeling. The bi - harmonic equation has excellent flexibility and can adapt to various types of boundary conditions. By reasonably setting these boundary conditions, it is ensured that the intermediate surface and the surrounding surface reach G1 continuity (i.e., the first - order derivative is continuous) at the splicing position. This method effectively avoids the discontinuous or sudden - change phenomena that may occur in traditional modeling methods, making the surface after modeling based on boundary information smoother and more natural.

[0016] (2) Reducing the computational complexity: Compared with the traditional boundary - information - based surface - modeling method, the PDE - based method in the present invention can effectively control the surface through a small number of parameters, boundary conditions, or right - hand sides, reducing the number of control points that need to be finely adjusted, thereby reducing the operation difficulty and computational complexity, and is especially suitable for processing complex geometric models. Description of the Drawings

[0017] Figure 1 is the flow - chart of the method of the present invention.

[0018] Figure 2 is the drawing of the surrounding surface of the automotive windshield. Among them, the surrounding surface is composed of four surface patches. In the azimuth order, these four surface patches are respectively called the upper surface, the lower surface, the left surface, and the right surface.

[0019] Figure 3 is the drawing of the surface after modeling based on boundary information, and the red part is called the intermediate surface. Among them, Figure (a) is the top view of the surface after modeling, and Figure (b) is the side view of the surface after modeling. Detailed Implementation Manner

[0020] The following further introduces the present invention in combination with specific embodiments.

[0021] An embodiment of the present invention provides a surface modeling method based on boundary information, including:

[0022] Step 1: Input the surrounding surface of the automotive windshield

[0023] Given the control point P at the i-th row and j-th column on the control mesh of the surrounding surface of the automotive windshield ij and its corresponding weight w ij , the order p in the direction (the row direction of the control mesh), the order q in the direction (the column direction of the control mesh); a} a (u a ≤ u a+1 , a = 0, 1,..., 2p + n + 1) (u a represents the a-th node value in the node vector U), V = {v b} b (v b ≤ v b+1 , b = 0, 1,..., 2q + m + 1) (v b represents the b-th node value in the node vector V); the degrees of the node vectors in the direction and the

[0024]

[0025] direction are n and m respectively, and construct the NURBS surface R(u, v) as the surrounding surface: The number of control points in the direction is p + n + 1, i p (u) is the i-th p-th B-spline basis function in the direction, is the

[0026] j-th q-th B-spline basis function in the

[0027] NURBS surface patch is called the surrounding surface.

[0028] NURBS surface patch is called the intermediate surface.

[0029] is a surface patch is the control point at the i-th row and j-th column on the control mesh of is the control point corresponding weight; and q are respectively direction and direction orders; direction and the knot vectors in the direction are respectively represents the c-th knot value in the knot vector ; represents the d-th knot value in the knot vector ; n, m respectively represent direction and the degrees of the knot vectors in the direction; q + m + 1 are respectively and the number of control points in the direction; is the i-th order B-spline basis function, is the j-th q-order B-spline basis function.

[0030] The surface patches R(u, v) and the surface patch are said to be G 1 continuous if and (This equation is called the G 1 continuity equation).

[0031] R(0, v) and respectively represent the control points corresponding to the surface patch R(u, v) and the surface patch when the parameter u takes zero and the parameter v takes the corresponding value. R u (0, v) and R v (0, v) respectively represent the control points corresponding to the surfaces obtained by taking the partial derivatives of the surface patch R(u, v) with respect to the parameters u and v when the parameter u takes zero and the parameter v takes the corresponding value; represents the control point corresponding to the surface obtained by taking the partial derivative of the surface patch with respect to the parameter u when the parameter u takes zero and the parameter v takes the corresponding value.

[0032] Without loss of generality, assume that the surrounding surface R(u, v) and the middle surface have the same knot vector V, that is α, β are functions of v, assume that the function α is a non-zero constant and β is 0.

[0033] To ensure G 0Continuity, the boundary conditions at the splicing joint are shared by the middle surface and the peripheral surface, which means that at the splicing joint, the boundary control points of the middle surface and the peripheral surface are exactly the same. This ensures the consistency of the geometric shape and topological structure at the splicing joint. Thus, the calculation formulas for the outermost control points and weights of the middle surface can be obtained:

[0034] Here, p 0j and are the control points at the j-th column of the 0-th row on the control meshes of the peripheral surface R(u, v) and the middle surface respectively, and w 0j and are the weights corresponding to p 0j and respectively.

[0035] At the same time, requirements are put forward for the continuity at its splicing joint. Here, it is required that the middle surface and the peripheral surface maintain G1 continuity. Under the G1 continuity requirement, there are no breakpoints at the splicing joint between the middle surface and the peripheral surface, and the tangent directions are the same. Using the derivative properties of NURBS surfaces, simplify the G1 continuity equation and derive the calculation formulas for the second outermost control points and weights of the middle surface at the splicing joint:

[0036]

[0037] Here and u p+1 are the (p + 1)-th knot values in the knot vector and the knot vector U respectively, p 1k and are the control points at the k-th column of the 1-st row on the control meshes of the surface patch R(u, v) and the surface patch respectively, w 1k and are the corresponding weights, and p 0k is the control point at the k-th column of the 0-th row on the control mesh of the surface patch R(u, v).

[0038] The outermost control points of the middle surface are exactly the same as the control points of the peripheral surface at the corresponding splicing position; the second outermost control points of the middle surface are derived from the G1 continuity equation. So far, the boundary control points (including the outermost and second outermost control points) of the middle surface have been obtained.

[0039] Step 3: Calculate the internal control points of the surface

[0040] To generate the internal control points (points other than the boundary control points) of the middle surface, a discrete format based on the biharmonic equation is used for solution.

[0041] The biharmonic equation is a fourth-order partial differential equation, which in two dimensions can be written as:

[0042]

[0043] u(x, y) is the unknown function of two variables to be solved, with x and y as independent variables;

[0044] f(x, y) is the given source term. In this problem, let f(x, y) = 0, that is, the homogeneous biharmonic equation is to be solved;

[0045] Ω is the solution domain, denotes the boundary of Ω;

[0046] The first and second boundary conditions for the above problem are given below.

[0047] First boundary condition: (That is, the value of u(x, y) on the boundary of Ω is the vector F). The sub-outer control points obtained in step 2 are distributed on the boundary of a specific rectangular grid. Set the three coordinate components of all points (i.e., the newly defined points) within this rectangular grid to 0. Integrate the sub-outer control points and the newly defined points obtained in step 2, and arrange these points in the order from left to right and from top to bottom to form the vector F. Therefore, each component of the vector F is a three-dimensional coordinate.

[0048] Second boundary condition: (That is, the directional derivative of u(x, y) with respect to the direction of γ is 0), where γ is the outer normal of the boundary of the middle surface.

[0049] According to the theory of existence and uniqueness of solutions of partial differential equations, it can be known that for the above homogeneous biharmonic equation under the defined first and second boundary conditions, the solution exists and is unique.

[0050] Next, the finite difference method will be used to solve the homogeneous biharmonic equation after defining the boundary conditions.

[0051] 3.1 Domain discretization

[0052] The solution domain Ω is divided into m 1 , m 2 equal parts along the x-direction (horizontal) and y-direction (vertical) respectively. Denote h 1 and h 2 as the step sizes in the x-direction and y-direction respectively. The grid points: (sh 1 , th 2 ) are denoted as (x s , y t ), where s = 0, 1,..., m 1 ; t = 0, 1,..., m 2, for a given set of \(s\) and \(t\), \((x s , y t ) represents the grid point at the \(s\)-th row and \(t\)-th column. Due to the correlation between the shape of the NURBS surface and the control points, the control points of the NURBS surface are taken as the unknowns for solving the biharmonic equation, that is, the unknown corresponding to the grid point \((x s , y t ) is \(P s,t \), representing the control point corresponding to the grid point at the \(s\)-th row and \(t\)-th column. Thus, the internal control points of the intermediate surface can be output. That is, at this time, the discrete grid in the solution domain \(\Omega\) has \((m 1 + 1)×(m 2 + 1)\) grid points, and the unknown corresponding to each grid point is the grid control point.

[0053] Arrange these grid control points in the order from left to right and from top to bottom to obtain the vector where \(P s,t \) is the control point with the value of the numerical solution of the homogeneous biharmonic equation at the point \((x s , y t ) as the coordinate components.

[0054] 3.2 Construction of the difference scheme

[0055] For regular interior points (all four adjacent grid points are in the solution domain \(\Omega\)): Replace the derivative with the difference quotient to obtain the following difference scheme with discrete solutions:

[0056]

[0057] If an equidistant partition is taken, that is, \(h 1 = h 2 \), the thirteen-point difference scheme of the homogeneous biharmonic equation can be simplified as follows:

[0058] \(P s,t-2 + 2P s-1,t-1 - 8P s,t-1 + 2P s+1,t-1 + P s-2,t - 8P s-1,t + 20P s,t - 8P s+1,t + P s+2,t + 2P s-1,t+1 - 8P s,t+1 + 2P s+1,t+1 + P s,t+2 = 0

[0059] The above difference scheme holds for \(s = 2, 3, \cdots, m 1 - 2, t = 2, 3, \cdots, m 2 - 2\).

[0060] For non-regular interior points: To have the same discrete error as the equation, extend one row outward for both the x-direction nodes and the y-direction nodes, i.e., s can be equal to -1 and m 1 +1, and t can be equal to -1 and m 2 +1. Using the condition that the left boundary derivative is 0, we have After simplification, we get P s,1 = P s,-1 , where s = 0, 1,..., m 1 . The treatment of the remaining boundary derivative conditions is similar.

[0061] Denote the vector Construct matrix A such that each row of matrix A can reflect the difference schemes for regular interior points and non-regular interior points.

[0062] Taking the regular interior point P s,t as an example, assume it is r components of the vector P * , where r = (s + 1)×(m 2 +1)+t+2. At the same time, thirteen points P that can establish a thirteen-point scheme with point P s,t (the subscripts s', t' represent the grid point at the s'-th row and t'-th column on the discrete grid of the solution region Ω corresponding to the control point P s',t' ). When s' = s, t' can take t - 2, t - 1, t, t + 1, t + 2; when s' = s - 1 or s + 1, t' can take t - 1, t + 1; when t' = t, s' can take s - 2, s - 1, s + 1, s + 2) The position e of these points in the vector P s',t' can be calculated according to e = (s' + 1)×(m * +1)+t'+2. 2

[0063] Then, the thirteen-point scheme for the regular interior point P s,t can be transformed into:[[]] The symbol "*" represents multiplication in the usual sense, A r,e represents the element in the r-th row and e-th column of matrix A, and r, e are calculated according to the above two equalities. If the above equality is expressed using the inner product, then A r ·P * = 0. This equality represents the thirteen-point difference scheme corresponding to the r-th component P * of the vector P s,t , where A r is the row vector corresponding to the r-th row of matrix A.

[0064] 3.3 Solving the linear equations

[0065] According to the difference schemes of all regular interior points and non-regular interior points, we can obtain (m 1 +3)×(m​2 +3) equations, combine all the equations to get the equation system AP * =G, where each component of G is a three-dimensional coordinate and is defined in the same way as vector F. This transforms the original problem into solving the linear equation system AP * =G, solving for vector P * , and extract the vector P * The points in the solution region Ω are integrated into a vector P, and each component of the vector P is the internal control point of the intermediate surface.

[0066] Step 4: Output the complete surface

[0067] The boundary control points obtained in step 2 and the internal control points obtained in step 3 are integrated together to form a complete control point matrix. According to the defined node vectors, the intermediate surface is generated using the NURBS surface generation algorithm, i.e., the surface corresponding to the car windshield. The surface generated in this way not only meets the G1 continuity requirements at the joints, ensuring a smooth visual transition, but also makes the surface of the car windshield modeled based on the boundary information look smooth and beautiful.

[0068] Example display

[0069] Step 1: Input the surrounding surface of the car windshield

[0070] Node vector

[0071] U=(u 0 ,u 1 ,...,u p ,u p+1 ,...,u p+n ,u p+n+1 ,...,u 2p+n+1 ),V=(v 0 ,v 1 ,...,v q ,v q+1 ,...,v q+m ,v q+m+1 ,...,v 2q+m+1 )in,

[0072] u 0 =...=u p =0,u p+n+1 =...=u 2p+n+1 =1,v 0 =...=v q =0,v q+m+1 =...=v 2q+m+1 =1 then p,q order

[0073] The NURBS surface R(u,v) is defined as follows:

[0074]

[0075] Given control points P ij and their corresponding weights w ij , the order of the basis functions p = q = 6, the degree n of the knot vector in one direction is 2, the degree m of the knot vector in the other direction is 2, and the knot vectors U = [0, 0, 0, 0, 0, 0, 0, 0.33, 0.66, 1, 1, 1, 1, 1, 1, 1], V = U. Construct the NURBS surface R(u, v) as the left surface among the four peripheral surfaces of the automotive windshield. The construction of the remaining three peripheral surfaces is similar, thus forming the complete peripheral surfaces of the automotive windshield( Figure 2 ), which are used to test the surface modeling method based on boundary information.

[0076] Step 2: Calculate the boundary information

[0077] The function α is set to 0.5. According to the boundary information of the peripheral surfaces and the calculation formula for the boundary control points of the intermediate surface, the boundary control points of the intermediate surface can be calculated.

[0078] Step 3: Calculate the internal control points of the surface

[0079] According to the result of Step 2 and the discrete difference format, a linear equation system AP * = G can be further obtained, and the vector P * is solved. Then extract the points in the vector P * that are included in the solution region Ω, integrate them into the vector P. Finally, the obtained vector P is a column vector with a length of 49, that is, there are 49 internal control points of the intermediate surface.

[0080] Step 4: Output the complete surface

[0081] Integrate the boundary control points obtained in Step 2 and the internal control points obtained in Step 3 to form a complete control point matrix. According to the knot vectors defined in Step 1, use the NURBS surface generation algorithm to generate the intermediate surface, that is, the surface corresponding to the automotive windshield. The intermediate surface and the peripheral surfaces together form the complete surface after modeling based on boundary information, as shown in Figure 3 (a), (b).

[0082] As can be seen from the numerical example, the present invention proposes a surface modeling method based on boundary information, which realizes surface modeling by calculating boundary information and solving the homogeneous biharmonic equation model. The high-order smoothness of the biharmonic equation not only ensures the overall smoothness and G1 continuity of the modeled surface, but also effectively avoids possible wrinkling or shape distortion problems. In addition, the parameters and the number of control points introduced by this technology are less, significantly reducing the computational complexity and making it perform excellently in dealing with complex geometric shapes. Therefore, when applied to the design of key components such as windshields and car doors in automobile manufacturing, the present invention can provide an efficient and reliable solution.

Claims

1. A surface modeling method based on boundary information, characterized in that: include: Step 1, input the peripheral surface of the automobile windshield: given the control mesh and corresponding weight of the peripheral surface of the automobile windshield, construct a NURBS surface as the peripheral surface; Step 2: Calculate boundary information: Based on the fact that the surrounding surface and the middle surface are G1 continuous, assume and Thus, the outermost control points and corresponding weights of the intermediate surface and the second outermost control points and corresponding weights of the intermediate surface at the joint are obtained; Where R(0,v) and Respectively represent the surface patch R(u,v) and the surface patch In the example, when parameter u is zero and parameter v is a corresponding value, the corresponding control point; R(u,v) is a patch on the surrounding surface with u and v as parameters. It is a patch on the intermediate surface with u and v as parameters; Step 3: Calculate the internal control points of the intermediate surface: Use a discrete format based on biharmonic equations to construct and solve the equation system AP * =G, we get vector P * , and extract the vector P * The points in the solution region Ω are integrated into a vector P, and each component of the vector P is the internal control point of the intermediate surface; Where A is a matrix constructed according to the difference format, P * To solve the grid points (x s ,y t ) corresponds to the unknown quantity P s,t The vector G is formed by integrating the sub-outer control points and the newly defined points, arranging these points in order from left to right and from top to bottom. The newly defined points are points in the rectangular grid formed by the sub-outer control points with all three coordinate components set to 0; Step 4: Output the complete surface: Integrate the boundary control points obtained in step 2 and the internal control points obtained in step 3 to form a complete control point matrix. According to the defined node vectors, use the NURBS surface generation algorithm to generate the intermediate surface, that is, the surface corresponding to the car windshield.

2. The surface modeling method based on boundary information according to claim 1, characterized in that: The NURBS surface constructed in step 1 is: Among them, u and v are parameters used to describe the position of points on the NURBS surface. i p (u)Yes The i-th p-th B-spline basis function in the direction, yes The j-th q-order B-spline basis function in the direction, w ij is the control point P in the i-th row and j-th column of the surrounding surface control grid ij The corresponding weight, p is the row direction of the control grid The order of q is the column direction of the control grid. The order of n and m are Direction and The degree of the direction node vector.

3. The surface modeling method based on boundary information according to claim 1, characterized in that: The outermost control points and corresponding weights of the mid-surface are: where p 0j and They are the peripheral surface R(u,v) and the intermediate surface The control point at row 0 and column j on the control grid is w 0j and Yes 0j and The corresponding weight.

4. The surface modeling method based on boundary information according to claim 1, characterized in that: The sub-outer control points and corresponding weights of the intermediate surface at the joint are: in and u p+1 are the p+1th node values ​​in the node vectors in the direction of the control grid row of the mid-surface and the peripheral surface, respectively. 1k and They are the peripheral surface R(u,v) and the intermediate surface The control point in row 1 and column k on the control grid of 1k and Yes 1k and The corresponding weight, p 0k is the control point in row 0 and column k on the control grid of the surrounding surface R(u,v); α is a non-zero constant, and q is the column direction of the control grid The order of n and m are Direction and The degree of the direction node vector.

5. The surface modeling method based on boundary information according to claim 1, characterized in that: where the vector P s,t is the grid point (x s ,y t ) corresponds to the unknown quantity, (x s ,y t ) represents the grid points in the sth row and tth column that divide the solution area Ω equally in the horizontal and vertical directions.

Images